Weighted dependency graphs Valentin Féray Institut für Mathematik, Universität Zürich Final conference of the MADACA project Domaine de Chalès, June 20th – June 24th 2016 V. Féray (UZH) Weighted dependency graphs Macada, 2016–06 1 / 26
Central limit theorems Theorem If Y 1 , Y 2 , . . . are independent identically distributed variables with finite variance, and X n = � n i = 1 Y i , then d X n − E ( X n ) → N ( 0 , 1 ) . (CLT) √ Var X n V. Féray (UZH) Weighted dependency graphs Macada, 2016–06 2 / 26
Central limit theorems Theorem If Y 1 , Y 2 , . . . are independent identically distributed variables with finite variance, and X n = � n i = 1 Y i , then d X n − E ( X n ) → N ( 0 , 1 ) . (CLT) √ Var X n Relax identical distribution hypothesis − → Lindeberg condition. V. Féray (UZH) Weighted dependency graphs Macada, 2016–06 2 / 26
Central limit theorems Theorem If Y 1 , Y 2 , . . . are independent identically distributed variables with finite variance, and X n = � n i = 1 Y i , then d X n − E ( X n ) → N ( 0 , 1 ) . (CLT) √ Var X n Relax identical distribution hypothesis − → Lindeberg condition. Relax independence hypothesis: leads to CLT for Markov chains, martingales, mixing sequences, exchangeable pairs, determinantal point processes, dependency graphs, . . . V. Féray (UZH) Weighted dependency graphs Macada, 2016–06 2 / 26
Central limit theorems Theorem If Y 1 , Y 2 , . . . are independent identically distributed variables with finite variance, and X n = � n i = 1 Y i , then d X n − E ( X n ) → N ( 0 , 1 ) . (CLT) √ Var X n Relax identical distribution hypothesis − → Lindeberg condition. Relax independence hypothesis: leads to CLT for Markov chains, martingales, mixing sequences, exchangeable pairs, determinantal point processes, dependency graphs, . . . Goal of the talk: give an extension of dependency graphs that has a wide range of application. V. Féray (UZH) Weighted dependency graphs Macada, 2016–06 2 / 26
Dependency graphs Dependency graphs (Petrovskaya/Leontovich, Janson, Baldi/Rinott, Mikhailov, 80’s) V. Féray (UZH) Weighted dependency graphs Macada, 2016–06 3 / 26
Dependency graphs A problem in random graphs 2 3 1 Erdős-Rényi model of random graphs G ( n , p ) : G has n vertices labelled 1,. . . , n ; 4 8 each edge { i , j } is taken independently with probability p ; 5 7 6 Example : n = 8 , p = 1 / 2 V. Féray (UZH) Weighted dependency graphs Macada, 2016–06 4 / 26
Dependency graphs A problem in random graphs 2 3 1 Erdős-Rényi model of random graphs G ( n , p ) : G has n vertices labelled 1,. . . , n ; 4 8 each edge { i , j } is taken independently with probability p ; 5 7 6 Example : n = 8 , p = 1 / 2 Question Fix p ∈ ( 0 ; 1 ) . Does the number of triangles T n satisfy a CLT? V. Féray (UZH) Weighted dependency graphs Macada, 2016–06 4 / 26
Dependency graphs A problem in random graphs 2 3 1 Erdős-Rényi model of random graphs G ( n , p ) : G has n vertices labelled 1,. . . , n ; 4 8 each edge { i , j } is taken independently with probability p ; 5 7 6 Example : n = 8 , p = 1 / 2 Question Fix p ∈ ( 0 ; 1 ) . Does the number of triangles T n satisfy a CLT? � � 1 if G contains the triangle ∆ ; T n = Y ∆ , where Y ∆ ( G ) = 0 otherwise. ∆= { i , j , k }⊂ [ n ] T n is a sum of mostly independent variables. V. Féray (UZH) Weighted dependency graphs Macada, 2016–06 4 / 26
Dependency graphs Dependency graphs Definition (Petrovskaya and Leontovich, 1982, Janson, 1988) A graph L with vertex set A is a dependency graph for the family { Y α , α ∈ A } if if A 1 and A 2 are disconnected subsets in L , then { Y α , α ∈ A 1 } and { Y α , α ∈ A 2 } are independent. Roughly: there is an edge between pairs of dependent random variables. V. Féray (UZH) Weighted dependency graphs Macada, 2016–06 5 / 26
Dependency graphs Dependency graphs Definition (Petrovskaya and Leontovich, 1982, Janson, 1988) A graph L with vertex set A is a dependency graph for the family { Y α , α ∈ A } if if A 1 and A 2 are disconnected subsets in L , then { Y α , α ∈ A 1 } and { Y α , α ∈ A 2 } are independent. Roughly: there is an edge between pairs of dependent random variables. Example � [ n ] � Consider G = G ( n , p ) . Let A = { ∆ ∈ } (set of potential triangles) and 3 { ∆ 1 , ∆ 2 } ∈ E L iff ∆ 1 and ∆ 2 share an edge in G . � [ n ] � Then L is a dependency graph for the family { Y ∆ , ∆ ∈ } . 3 V. Féray (UZH) Weighted dependency graphs Macada, 2016–06 5 / 26
Dependency graphs Dependency graphs Definition (Petrovskaya and Leontovich, 1982, Janson, 1988) A graph L with vertex set A is a dependency graph for the family { Y α , α ∈ A } if if A 1 and A 2 are disconnected subsets in L , then { Y α , α ∈ A 1 } and { Y α , α ∈ A 2 } are independent. Roughly: there is an edge between pairs of dependent random variables. ✞ ☎ Note: L has degree O ( n ) Example ✝ ✆ � [ n ] � Consider G = G ( n , p ) . Let A = { ∆ ∈ } (set of potential triangles) and 3 { ∆ 1 , ∆ 2 } ∈ E L iff ∆ 1 and ∆ 2 share an edge in G . � [ n ] � Then L is a dependency graph for the family { Y ∆ , ∆ ∈ } . 3 V. Féray (UZH) Weighted dependency graphs Macada, 2016–06 5 / 26
Dependency graphs Janson’s normality criterion Setting: for each n , { Y n , i , 1 ≤ i ≤ N n } is a family of bounded random variables; | Y n , i | < M a.s. we have a dependency graph L n with maximal degree ∆ n − 1. we set X n = � N n i = 1 Y n , i and σ 2 n = Var ( X n ) . V. Féray (UZH) Weighted dependency graphs Macada, 2016–06 6 / 26
Dependency graphs Janson’s normality criterion Setting: for each n , { Y n , i , 1 ≤ i ≤ N n } is a family of bounded random variables; | Y n , i | < M a.s. we have a dependency graph L n with maximal degree ∆ n − 1. we set X n = � N n i = 1 Y n , i and σ 2 n = Var ( X n ) . Theorem (Janson, 1988) � � 1 / s ∆ n N n σ n → 0 for some integer s . Then X n satisfies a CLT. Assume that ∆ n V. Féray (UZH) Weighted dependency graphs Macada, 2016–06 6 / 26
Dependency graphs Janson’s normality criterion Setting: for each n , { Y n , i , 1 ≤ i ≤ N n } is a family of bounded random variables; | Y n , i | < M a.s. we have a dependency graph L n with maximal degree ∆ n − 1. we set X n = � N n i = 1 Y n , i and σ 2 n = Var ( X n ) . Theorem (Janson, 1988) � � 1 / s ∆ n N n σ n → 0 for some integer s . Then X n satisfies a CLT. Assume that ∆ n � n � , ∆ n = O ( n ) , while σ n ≍ n 2 . (for fixed p ) For triangles, N n = 3 Corollary Fix p in ( 0 , 1 ) . Then T n satisfies a CLT. (also true for p n → 0 with np n → ∞ ; originally proved by Rucinski, 1988). V. Féray (UZH) Weighted dependency graphs Macada, 2016–06 6 / 26
Dependency graphs Applications of dependency graphs to CLT results mathematical modelization of cell populations (Petrovskaya, Leontovich, 82); subgraph counts in random graphs (Janson, Baldi, Rinott, Penrose, 88, 89, 95, 03); Geometric probability (Avram, Bertsimas, Penrose, Yukich, Bárány, Vu, 93, 05 , 07); pattern occurrences in random permutations (Bóna, Janson, Hitchenko, Nakamura, Zeilberger, 07, 09, 14). m -dependence (Hoeffding, Robbins, 53, . . . ; now widely used in statistics) is a special case. (Some of these applications use variants of Janson’s normality criterion, which are more technical to state and omitted here. . . ) V. Féray (UZH) Weighted dependency graphs Macada, 2016–06 7 / 26
Dependency graphs Not an application of dependency graphs Random graph G ( n , M ) : 2 G has n vertices labelled 1,. . . , n ; 3 1 The edge-set of G is taken uniformly 4 8 among all possible edge-sets of cardinality M . 5 7 Example with n = 8 and M = 14 6 V. Féray (UZH) Weighted dependency graphs Macada, 2016–06 8 / 26
Dependency graphs Not an application of dependency graphs Random graph G ( n , M ) : 2 G has n vertices labelled 1,. . . , n ; 3 1 The edge-set of G is taken uniformly 4 8 among all possible edge-sets of cardinality M . 5 7 Example with n = 8 and M = 14 6 � n � If p = M / , each edge appears with probability p , but no independence 2 any more! V. Féray (UZH) Weighted dependency graphs Macada, 2016–06 8 / 26
Dependency graphs Not an application of dependency graphs Random graph G ( n , M ) : 2 G has n vertices labelled 1,. . . , n ; 3 1 The edge-set of G is taken uniformly 4 8 among all possible edge-sets of cardinality M . 5 7 Example with n = 8 and M = 14 6 � n � If p = M / , each edge appears with probability p , but no independence 2 any more! Question � n � Fix p ∈ ( 0 ; 1 ) and M = p . Does the number of triangles T n in 2 G ( n , M n ) satisfy a CLT? V. Féray (UZH) Weighted dependency graphs Macada, 2016–06 8 / 26
Recommend
More recommend
